Theoretical Seismology 2: Wave Propagation Thailand Training Program in Seismology and Tsunami Warnings, May 2006 Theoretical Seismology 2: Wave Propagation Seismic waves in an elastic medium ・ Rays and Ray Paths How they propagate in the Earth Travel-time curves ・ Near-Field Terms (Static Displacements) • Far-Field Terms (P, S, Surface waves) • Surface Waves ・ Normal modes (Free oscillations of the Earth) Explosion P waves Psychology test Homogeneous Earth (seismologist) (engineer) (tsunamicist?) Ray Paths in a Layered Medium Snell’s law: sin a1 / sin a2 = a1 / a2 a1 q1 Slower a1 q1 Faster q2 Faster q2 Slower a２ a２ a1 < a2 a1 > a2 Travel-time Curves for Travel-time curves Ray Paths in a Layered Medium Time 1/a3 1/a2 1/a1 Distance (D=T/velocity) a1 head wave a2 a3 Velocity structure of the Earth Ray Paths in a Gradient Travel-time curves Time Velocity gradient can be treated as a series of thin homogeneous layers. 1/a3 1/a2 1/a1 Distance (D=T/velocity) a1 a2 a3 Moho Andrija Mohorovicic (1857-1936) Found seismic discontinuity at 30 km depth in the Kupa Valley (Croatia). Mohorovicic discontinuity or ‘Moho’ Boundary between crust and mantle Structure in the Earth Conrad and Moho Discontinuities Low velocity zone Forward Branch Receding Branch Forward Branch Shadow Zone Forward Branch (PKPbc) Receding Branch (PKPab) PcP Receding Branch A Forward PKP Branch B C Forward Branch Shadow PcP Shadow Zone P Zone Forward Branch Receding Branch Forward Branch Not shown: PKP(DEF) and PKiKP Other notation for core phases: ABC branch known as P2l DEF branch known as P1ll PKP(DEF) known as PKIKP Point B is a caustic PcP Core Reflections Faulting Seismic waves Other aspects of wave propagation: • Diffracted Waves ・ Surface Waves ・ Static Displacements ・ Frequency content • Normal Modes Other aspects of wave propagation • Diffracted Waves ・ Surface Waves ・ Static Displacements ・ Frequency content and wavelength • Normal Modes 1-D Wave Equation u1 1 u1 2 2 x12 c t 2 1-D wave equation c = propagation speed 2 u1 1 2 u1 x1 2 c t 2 Solution u( x, t ) A sin[ (t x / c)] 2 T = wave period T = angular frequency LW 3.2.1 Wave Period and Wavelength Velocity = Wavelength / Period Space x Velocity 6 km/s wavelength period 50 s Wavelength 300 km Time t period 50 s frequency = 1/period= 0.02 hz period Period Wavelength Body waves 0.1 to 50 sec 50 m to 500 km （P・S） Surface waves 10 to 350 sec 30 to 1000 km Free Oscillations 350 to 3600 sec 1000 to 10000 km Static Displacements - Other aspects of wave propagation •Diffracted waves ・ Surface waves ・ Static Displacements (amplitude at zero frequency) ・ Frequency content • Normal modes 3-D Wave Equation with Source 2u 2 f ( 2 )( u ) ( u ) t source spatial 2nd derivative Near-field Terms (Static Displacements) Solution 1 1 r/ 1 1 r 1 1 r u ( x, t ) a M 0 (t )d A IP M 0 (t ) A IS 2 M 0 (t ) N A 4 4 r r/ 4a 2 r2 a 4 2 r 1 1 r 1 1 r A FP M 0 (t ) A FS M 0 (t ) 4a 3 r a 4 3 r Far-field Terms (P, S Waves) Near-field terms ・ Static displacements r/a r/ ・ Only significant close to the fault ・ Source of tsunamis r/a r/ t → Static Displacements Bei-Fung Bridge near Fung-Yan city, 1999 Chi-Chi, Taiwan earthquake Static displacements Co-seismic deformation of 2003 Tokachi-oki Earthquake (M8.0) Generation of Tsunami from Near-field Term EA PAC Far-field Terms 1 A FP 1 r M 0 (t ) 1 1 r A FS M 0 (t ) 4a 3 r a 4 3 r ・ Propagating Waves ・ No net displacement in an elastic medium ・ P waves ・ S waves Other aspects of wave propagation • Diffracted Waves ・ Surface Waves ・ Static Displacements ・ Frequency content • Normal Modes Surface Waves Group Velocity (km/sec) Love Rayleigh S Period (sec) Shearer, Fig. 8.1 January 26, 2001 Gujarat, India Earthquake (Mw7.7) Body waves vertical Rayleigh Waves P PP S SS radial transverse Love Waves Recorded in Japan at a distance of 57o (6300 km) Other aspects of wave propagation • Diffracted Waves ・ Surface Waves ・ Static Displacements ・ Frequency content • Normal Modes Free Oscillations of the Earth (Normal Modes) Few minutes after the earthquake Few hours after the earthquake (0S20) Constructive interferences free oscillations (or stationary waves) Standing Waves with Periods < 54 min, amplitudes < 1 mm Observable months after great earthquakes (e.g. Sumatra, Dec 2004) From Michel van Camp, Royal Obs. of Belgium Normal Modes (Stein and Gellar 1978) Free Oscillations of the Earth (Daishinji, Fukui Prefecture) 1960 Chile Earthquake Useful for studies of ・ Interior of the Earth ・ Largest earthquakes Toroidal and Spheroidal Modes Toroidal Spheroidal Dahlen and Tromp Fig. 8.5, 8.17 Natural Vibrations of the Earth Indexes describe spherical harmonics Shearer Ch.8.6 Lay and Wallace, Ch. 4.6 Free Oscillations l=1 m=1 Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html Free Oscillations l=1 m=2 Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html Free Oscillations l=1 m=3 Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html Structure: Free Surface Earth is a not homogenous whole-space Free surface causes many complications - surface waves - reflections (pP, sP, sS) Summary Rays Velocity structure includes gradients, discontinuities and LVZ’s, causing complicated ray paths through the Earth (P, PKP, PcP) Wave theory explains ・ P and S waves ・ Static displacements ・ Surface waves Normal Modes The Earth rings like a bell at long periods Why are observed seismograms so messy ?
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